From what I've studied, it seems like ACW organization and tactics aren't very similar and "Napoleonic tactics against modern weaponry" seems to be a bit of a popular misconception or meme (i.e. like "Spartans were the best warriors in all of history!"), or at least an overstatement. Or I suppose "Napoleonic tactics" is just used as layman shorthand for "soldiers shooting at each other in tightly packed lines" without a full appreciation of how nuanced Napoleonic combat could actually be. Here are some of my specific points so you can specifically address what/why I get right and wrong:

• As I've read in Jac Weller's Wellington at Waterloo, French army corps were intended to be combined-arms formations capable of acting independently but also together, with the overall army commander (i.e. Napoleon himself) having direct commander over more specialized assets (such as an artillery or cavalry corps) to reinforce their subordinates. This style of organization was partially copied by the Prussians; their armies used corps, but they tended to act like separate armies than parts of an army that could act independently. I haven't read anything of this kind of thing in ACW battles: organization seems a lot more haphazard with brigades, regiments, divisions, and so on all mixed together. While corps exist, they seem to be almost all infantry with some supporting artillery if available, rather than an organic combined-arms formation.
• In addition to combined-arms organization, I read in John Keegan's The Face of Battle that although infantry was the primary force of Napoleonic armies, there was also some emphasis on combined-arms tactics. For example, the approach of cavalry would force infantry into defensive squares, after which mobile horse artillery would be brought up to blast holes in the squares.
• Going off the idea that ACW armies aren't combined-arms to the extent Napoleonic ones were, I read in Alexander Bevin's How the South Could Have Won the Civil War that cavalry and artillery played less of a role in ACW armies than Napoleonic ones, mostly due to the enhanced range of infantry rifles over muskets. He argues that the range of the Minie-ball rifle outranged canister shot by artillery, precluding the effective Napoleonic tactic of rolling up cannons just outside of enemy musketry and blasting them with canister (John Keegan mentions this tactic as well, when he talks about mobile horse artillery accompanying cavalry charges). He also makes the argument that rifled cannons and explosive shells weren't advanced enough to inflict much damage at long range either. Likewise, he uses the enhanced range of rifles to say that since infantry could shoot at cavalry at much longer ranges, cavalry's main battlefield role was obsolete.
• With regards to infantry specifically, there seems to be far less variety of infantry formations and tactics in the ACW than Napoleonic era. With heavy cavalry gone from the battlefield, I've never read of infantry squares in any ACW battle. Likewise, I get the impression that ACW infantry fought primarily in line formation, since French-style deep columns have never been mentioned in any book/video/presentation I've ever seen on the ACW. I am aware of some skirmishing incidents (such as the First Day of Gettysburg), but ACW skirmishes seem to be preliminary or independent actions, as opposed to the Napoleonic era where skirmishing is a constant component of set-piece battles. Jac Weller argued that French skirmishers (and deep columns) were essential to the Revolution's early victories, and that the European nations copied the French with specially trained light companies in their regiments or independent units like the 95th Rifles. They would screen the main infantry forces, beat back the enemy light infantry, and harass enemy formations all throughout the battle. I haven't heard of such military concepts in the ACW.
• Lastly, the ACW has a much greater focus on fortifications and trenches in battles. This is opposed to the Napoleonic era, in which field fortifications were looked upon with suspicion at best. Bevin quotes Napoleon himself as hating them for disorganizing troops and making men too afraid to leave, while Weller describes Wellington as not a fan either, since static defenses pinned troops in place whereas Wellington's style of defense was based on the ability to maneuver reserves and launch counterattacks.